2.6: Second Order Phase-Locked Loops
2.6 Second Order Phase-Locked Loops
The first order loop analysis for the three different inputs suggests a general equation for the loop filter,
The variable n, represents the desired order of the phase-locked loop. Jaffe and Rechtin  investigated the optimum loop filters for phase-locked loops for different inputs to the phase-locked loop. Their approach is similar to Weiner filter theory, and for a frequency step input, the optimum filter is found to have the form of the active lead-lag filter discussed below.
The first order loop failed with an input response ?( t) = a t, so to provide a matched response to this particular input, we would like a term corresponding to a t. From Equation 2-57, a second order loop requires a loop filter of the form f ( t) = c 0 ?( t) + c 1. The Laplace Transform of this filter is
With the appropriate substitutions, this filter can be rewritten in the form
Three traditional filters for a second order loop are shown in Figure 2.5. Note the active loop filter is identical to Equation 2-59, where we attempted to match the filter's response to the phase input. Any of the filters yields a second order loop, although the active lead-lag filter provides superior performance.
Figure 2.5: Analog Loop Filters 
The second order control loop is distinguished by the appearance of a second-degree polynomial in the denominator of Equation 2-48. However, specifying...